of Edinburgh, Session 1873-74. 233 
tinued fraction are expressible in an unexpectedly simple and 
elegant manner. Thus — 
and thus by induction we prove that — 
a 2 + a 3 + 
15. In virtue of this connection continuants will be found of the 
utmost aid in investigating the properties of continued fractions. 
The following are a few instances of this relating to those con- 
tinued fractions which are expressible in the form of quadratic 
surds. 
16. Consider the periodic continued fraction — 
K-i 
* • • 4" «»_ 1 + 
( h \ \ 
' \a, .a . -i, aj 
k ( h 2 h "~ l ) 
\a,,a 3 . .. o._, , aj 
(XIII.) 
a + : 
«i + (l 2 + d 3 -f 
+ a 2 + a x + 2A + , 
where the asterisks are used like the superposed dots in the nota- 
tion of decimal fractions to indicate the recurring portion or 
period. 
